technical analysis compared to mathematical - sophia antipolis

Journal of Banking & Finance 31 (2007) 1351–1373

www.elsevier.com/locate/jbf

Technical analysis compared to mathematicalmodels based methods under parameters

mis-specification

Christophette Blanchet-Scalliet a,*, Awa Diop b, Rajna Gibson c,Denis Talay b, Etienne Tanre b

a Laboratoire Dieudonne, Universite Nice Sophia-Antipolis, Parc Valrose 06108 Nice Cedex 2, Franceb INRIA, Projet OMEGA, 2004 Route des Lucioles, BP93, 06902 Sophia-Antipolis, France

c NCCR FINRISK, Swiss Banking Institute, University of Zurich, Plattenstrasse 14, Zurich 8032, Switzerland

Available online 11 December 2006

Abstract

In this study, we compare the performance of trading strategies based on possibly mis-specifiedmathematical models with a trading strategy based on a technical trading rule. In both cases, the traderattempts to predict a change in the drift of the stock return occurring at an unknown time. We explicitlycompute the trader’s expected logarithmic utility of wealth for the various trading strategies. We nextrely on Monte Carlo numerical experiments to compare their performance. The simulations show thatunder parameter mis-specification, the technical analysis technique out-performs the optimal alloca-tion strategy but not the Model and Detect strategies. The latter strategies dominance is confirmedunder parameter mis-specification as long as the two stock returns’ drifts are high in absolute terms.� 2006 Elsevier B.V. All rights reserved.

MSC: 60G35; 93E20; 91B28; 91B26; 91B70

JEL classification: G11; G14; C15; C65

Keywords: Stochastic models; Model specification; Portfolio allocation; Chartist

0378-4266/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.jbankfin.2006.10.017

* Corresponding author. Tel.: +33 4 92 07 64 89; fax: +33 4 93 51 79 74.E-mail addresses: [email protected] (C. Blanchet-Scalliet), [email protected] (A. Diop),

[email protected] (R. Gibson), [email protected] (D. Talay), [email protected](E. Tanre).

mailto:[email protected]





1352 C. Blanchet-Scalliet et al. / Journal of Banking & Finance 31 (2007) 1351–1373

1. Introduction

The financial services industry typically relies on three main approaches to make invest-ment decisions: the fundamental approach that uses fundamental economic principles toform portfolios, the technical analysis approach that uses price and/or volume historiesand the mathematical approach that is based on mathematical models. Technical analysishas been used by professional investors for more than a century. The academic communityhas looked at its foundations and its performance with a rather skeptical frame of mind.Indeed, technical analysis techniques have limited theoretical justification, and they standin contradiction to the conclusions of the efficient market hypothesis. More recently, therehas been a renewal of academic interest in the performance of technical analysis basedmethods. Indeed, the pioneering study by Brock et al. (1992) applied 26 trading rules tothe Dow Jones Industrial Average and found that they significantly out-perform a bench-mark of holding cash. In their impressive study, Sullivan et al. (1999) examine close to8000 technical trading rules and repeat Brock et al. study while correcting it for datasnooping problems. They find that the trading rules examined by Brock et al. do notgenerate superior performance out-of-sample. Lo et al. (2000) propose to use a non-para-metric kernel regressions pattern recognition method in order to automate the evaluationof technical analysis trading techniques. In their comprehensive study they compare theunconditional and the conditional – on technical analysis indicators – distribution of alarge number of stocks traded on the NYSE/AMEX and on the NASDAQ. They concludethat ‘‘several technical indicators do provide some incremental information and may havesome practical value’’. However, as pointed out by Jegadeesh, 2000 in his comment of theLo et al. (2000) paper, none of the technical analysis indicators examined by the authors isable to identify profitable investment opportunities. Thus, it seems that the debate aboutthe effectiveness of technical analysis usefulness is still very much alive.

The purpose of our study is to examine chartist and mathematical models based tradingstrategies by providing a conceptual framework where their performance can be com-pared. If one considers a non-stationary economy, it is impossible to specify and calibratemathematical models that can capture all the sources of parameter instability during along time interval. In such an environment, one can only attempt to divide any long invest-ment period into sub-periods such that, in each of these sub-periods, the financial assetsprices can reasonably be supposed to follow some particular distribution (e.g., a stochasticdifferential system with a fixed volatility function). Due to the investment opportunity set’sinstability, each sub-period must be short. Therefore, one can only use small amounts ofdata during each sub-period to calibrate the model, and the calibration errors can be sub-stantial. Yet, any investment strategy’s performance depends on the underlying modelcharacterizing the evolution of the investment opportunity set and also on the parametersinvolved in the model. Thus, in a non-stationary economy, one can use strategies whichhave been optimally designed under the assumption that the market is well described bya prescribed model, but these strategies can be extremely misleading in practice becausethe prescribed model does not fit the actual evolution of the investment opportunity set.In such a situation, is one better of using a technical analysis based trading rule whichis free of any model dependency? In order to answer that question one should comparethe performance obtained by using erroneously calibrated mathematical models withthe one associated with technical analysis techniques. To our knowledge, this questionhas not yet been investigated in the academic literature.

C. Blanchet-Scalliet et al. / Journal of Banking & Finance 31 (2007) 1351–1373 1353

More specifically, we here consider the following test case: the agent in a frictionlesscontinuous-time economy can invest in a riskless asset and in a stock. The instantaneousexpected rate of return of the stock changes once at an unknown random time. We com-pare the performance of traders who respectively use:

• A technical analysis technique, namely the simple moving average technique in order topredict the change in the stock returns’ drift.

• A portfolio allocation strategy which is optimal when the mathematical model is per-fectly specified and calibrated.

• Two mathematical strategies called ‘‘Model and Detect’’ strategies aimed at detectingthe time of the drift change.

• The three previous strategies under mis-specified parameters (due to the error oncalibration).

The study is divided into two parts: a mathematical part which, whenever possible, pro-vides analytical formulae for portfolios managed by means of mathematical and technicalanalysis strategies and a numerical part which provides comparisons between the variousstrategies’ performance. Based on the numerical simulations, we find that the chartiststrategy can out-perform optimal portfolio allocation models when there is parametermis-specification. However, the ‘‘Model and Detect mathematical strategies’’ clearly dom-inate the chartist trading rule even when they are subject to parameter mis-specification.

The paper is organized as follows: In Section 2, we describe the basic setting underlyingour mathematical modeling. In Sections 3 and 4, we examine the performance of a traderwhose strategy is based on mathematical models. In Section 3, we examine the optimalportfolio allocation strategy. We give explicit formulas for the optimal wealth and theportfolio strategy of a trader who perfectly knows all the parameters characterizing theinvestment opportunity set and thus fully describe the best financial performance thatone can expect within our model. In Section 4, we consider a trader who uses mathemat-ical models in order to detect the change time s in the drift of the stock price process asearly and reliably as possible: he/she selects a stopping time H* adapted to the filtrationgenerated by (St), which serves as an ‘‘alarm signal’’ (this strategy is called ‘‘Model andDetect’’). In Section 5, we consider the performances of the optimal portfolio allocationstrategy and of the Model and Detect strategy when the trader mis-specifies the parametersof the model. In Section 6, we focus on a technical analyst who uses a simple moving aver-age indicator to detect the time at which the drift of the stock return switches. We char-acterize his/her expected utility of wealth in the logarithmic case. We also numericallyillustrate the properties of his/her strategy’s performance. Finally, in Section 7, we com-pare the performances of the mis-specified mathematical strategies to those of the technicalanalysis technique.1

2. Description of the setting

We consider a frictionless continuous-time economy with two assets that aretraded continuously. The first one is an asset without systematic risk, typically a

1 A short version of this paper has been published in Blanchet-Scalliet et al. (2006).


risk-less bond (or bank account), whose price at time t evolves according to the followingequation

dS0t ¼ S0

t r dt;

S00 ¼ 1:

(

The second asset is a stock subject to systematic risk. We model the evolution of its price attime t by the linear stochastic differential equation

dSt ¼ St l2 þ ðl1 � l2Þ1ðt6sÞ� �

dt þ rSt dBt;

S0 ¼ S0;

(ð2:1Þ

where (Bt)06t6T is a one-dimensional Brownian motion on a given probability spaceðX;F;PÞ. The random time of the stock return drift change s is independent of B andhas an exponential distribution with parameter k:

Pðs > tÞ ¼ e�kt; t P 0: ð2:2Þ

At time s, which is neither known, nor directly observable, the instantaneous expected rateof return changes from l1 to l2.

We suppose that the parameters l1, l2, r > 0 and r P 0 satisfy

l1 �r2

2< r < l2 �

r2

2:

The main purpose of this study will be to use this setting in order to examine if math-ematical models used to detect the time change in the drift of the stock price process leadto better predictions and thus to superior performance than a very popular technical anal-ysis method based on a simple moving average signal.

3. The optimal portfolio allocation strategy under a change of drift

3.1. Introduction

We start by characterizing the optimal wealth and portfolio allocation of a trader whoperfectly knows all the parameters l1, l2, k, r and r. Of course, this situation is unrealistic.However, it is worth computing the best performance that one can expect within our set-ting. This performance represents an optimal benchmark for mis-specified allocation strat-egies relying either on a mathematical model or on technical analysis.

Let pt be the proportion of the trader’s wealth invested in the stock at time t; theremaining proportion 1 � pt is invested in the bond. For a given, non random, initial cap-ital x > 0, let W x;p

� denote the wealth process corresponding to the portfolio (pÆ). We have

dW x;pt ¼ W x;p

t ðr dt þ pt½ðl1 � r þ ðl2 � l1Þ1ðs6tÞÞdt þ rdBt�Þ: ð3:1Þ

Now, let U(Æ) denote a utility function. We suppose that U is strictly increasing, concave, ofclass C2(0,1) and satisfies

U 0ð0þÞ ¼ limx#0

U 0ðxÞ ¼ 1; U 0ð1Þ ¼ limx"1

U 0ðxÞ ¼ 0:


Let AðxÞ denote the set of admissible strategies, that is, the set of processes p which takevalues in [0,1] and are progressively measurable with respect to the filtration FS generatedby the observed prices St.

It is easy to see that, for all process p in AðxÞ, W x;pt > 0 for all t.

The investor’s objective is to maximize his/her expected utility of wealth at the terminaldate T. He/she solves the following optimization problem:

V ðxÞ :¼ supp�2AðxÞ

E½UðW pTÞjW p

0 ¼ x�: ð3:2Þ

In order to compare the performance of the optimal strategy defined by (3.2) to the onepursued by a technical analyst, we impose constraints on the portfolio weights. Indeed, wewill below assume that the technical analyst invests all of his/her wealth either in the stockor in the bond depending upon the moving average signal. We therefore assume that theportfolio weights of the trader pursuing an optimal strategy are also constrained to liewithin the interval [0,1] in the absence of short selling.

3.2. The case of general utility functions

In order to compute the constrained optimal wealth of the trader, we use the martingaleapproach to stochastic control problems as developed by Karatzas, Shreve, Cvitanic, etc.More precisely, we follow and carefully adapt the martingale approach to the well knownoptimal consumption-portfolio problem studied by Merton (1971). We emphasize that ourtrader’s situation differs from the Merton problem for the following three reasons:

• The drift coefficient of the dynamics of the risky asset return is not constant over time(since it changes at the random time s).

• We face some subtle measurability issues since the trader’s strategy needs to be adaptedwith respect to the filtration generated by (St). As already mentioned, the drift changeat the random time s makes this filtration different from the filtration generated by theBrownian motion (Bt).

• The portfolio weight p is constrained to lie in a finite interval ([0,1]).

These three features of our problem make it hard to construct optimal strategies.Indeed, in the case of a general utility function, we need an additional hypothesis describedbelow to prove the existence of an optimal constrained allocation strategy p*, and to exhi-bit an abstract representation for the corresponding optimal portfolio process.

As in Karatzas and Shreve (1998), we introduce an auxiliary unconstrained market Mm

defined as follows: Let D be the subset of fFSt g – progressively measurable processes

m : ½0; T � � X! R such that

E

Z T

0

mðtÞ� dt <1; where mðtÞ� :¼ � infð0; mðtÞÞ:

The bond price process S0(m) and the stock price S(m) satisfy

dS0t ðmÞ

S0t ðmÞ

¼ dS0t

S0t

þ m�ðtÞdt;

dStðmÞStðmÞ

¼ dSt

Stþ ðmðtÞ þ m�ðtÞÞdt:


We compute the optimal allocation strategy for each auxiliary unconstrained market dri-ven by a process m (see Proposition 3.1). We conclude with Proposition 3.2 which links theoptimal strategy for the constrained problem with the set of optimal strategies for auxil-iary unconstrained markets.

For each auxiliary unconstrained market, let Aðx; mÞ denote the set of admissible strat-egies, that is,

Aðx; mÞ :¼ fp� –FSt –progressively measurable process s:t:

W m;p0 ¼ x; W m;p

t > 0 for all t > 0g:We have to solve the following problem Pm:

V ðm; xÞ :¼ supp�2Aðm;xÞ

E½UðW m;pT ÞjW

m;p0 ¼ x�;

where

dW m;pt

W m;pt

¼ ptdStðmÞStðmÞ

þ ð1� ptÞdS0

t ðmÞS0

t ðmÞ¼ dW p

t

W pt

þ ðm�ðtÞ þ ptmðtÞÞdt: ð3:3Þ

To characterize optimal allocation strategies and their associated wealth level, we need tointroduce four processes which are adapted to the filtration FS generated by the observedprice process.

• The exponential likelihood-ratio process Lt is defined by

Lt ¼St

S0

� �l2�l1r2

exp � 1

2r2ðl2 � l1Þ

2 þ 2ðl2 � l1Þ l1 �r2

2

� �� t

� �: ð3:4Þ

• The conditional a posteriori probability Ft that the change point has appeared beforetime t is F t :¼ Pðs 6 tjFS

t Þ:

F t ¼kektLt

R t0

e�ksL�1s ds

1þ kektLt

R t0

e�ksL�1s ds

ð3:5Þ

• The innovation process B is defined by

Bt ¼1

rlogðStÞ � l1 �

r2

2

� �t � ðl2 � l1Þ

Z t

0

F s ds� �

; t P 0: ð3:6Þ

The process B is a Brownian motion for the filtration FS .• The exponential process H m

t :

H mt ¼ exp �

Z t

0

l1� rþ mðsÞr

þðl2�l1ÞF s

r

� �dBs�

1

2

Z t

0

l1� rþ mðsÞr

þðl2�l1ÞF s

r

� �2

ds

!:

Proposition 3.1. For each m 2 D, the optimal wealth is

W m;�T ¼ ðU 0Þ

�1 yH mTe�rT�

R T

0m�ðtÞ dt

� �

W m;�t ¼

ertþR t

0m�ðsÞ ds

H mt

E H mTe�rT�

R T

0m�ðsÞ dsðU 0Þ�1 yH m

Te�rT�

R T

0m�ðsÞ ds

� �jFS

t

� �:


Moreover, the optimal strategy satisfies

pm;�t ¼ r�1 l1 � r þ ðl2 � l1ÞF t þ mðtÞ

rþ /t

H mt W

m;�t e�rt�R t

0m�ðsÞ ds

0@ 1A; ð3:7Þ

where Ft is defined as in (3.5), y stands for the Lagrange multiplier, that is, such that

E H mT exp �rT �

Z T

0

m�ðtÞdt� �

ðU 0Þ�1 yH mT exp �rT�

Z T

0

m�ðtÞdt� ��

¼ x:

and / is a FSt adapted process which satisfies

E H mTe�rT�

R T

0m�ðtÞ dtðU 0Þ�1 yH m

Te�rT�

R T

0m�ðtÞ dt

� �FS

t

�¼ xþ

Z t

0

/sdBs:

�The proof is postponed to Appendix.

In view of (3.3), we observe that for each constrained strategy p, m�(t) + ptm(t) P 0, andtherefore W m;p

t P W pt and V(m, x) P V(x). So, V ðxÞ 6 inf m2DV ðm; xÞ. The following propo-

sition tells us that, if the minimum is attained, then the minimizing auxiliary strategy emprovides the optimal constrained strategy p*.

Proposition 3.2. If there exists em in D such that

V ðem; xÞ ¼ infm2D

V ðm; xÞ; ð3:8Þ

then an optimal portfolio p�;~m for the unconstrained problem P~m is also an optimal portfolio for

the constrained original problem P, such that

W �t ¼ W p�;~m;em

t and V ðxÞ ¼ V ðem; xÞ: ð3:9ÞAn optimal portfolio allocation strategy is

p�t :¼ r�1 l1 � r þ ðl2 � l1ÞF t þ emðtÞr

þ /t

Hemt W �t e�rt�R t

0em�ðsÞ ds

0@ 1A; ð3:10Þ

where Ft is defined as in (3.5).

Proof. See the proof in Karatzas and Shreve (1998, p. 275). h

3.3. The case of the logarithmic utility function

When the agent has a logarithmic utility function, we can verify the existence of a em sat-isfying (3.8) and explicit the optimal allocation strategy p* and its associated wealth W*,x.

Proposition 3.3. If U(Æ) = log(Æ) and the initial endowment is x, then the optimal wealth

process and strategy are

W �;xt ¼

x expðrt þR t

0em�ðsÞdsÞ

Hemt ; ð3:11Þ


where

emðtÞ ¼�ðl1 � r þ ðl2 � l1ÞF tÞ if

l1 � r þ ðl2 � l1ÞF t

r2< 0;

0 ifl1 � r þ ðl2 � l1ÞF t

r22 ½0; 1�;

r2 � ðl1 � r þ ðl2 � l1ÞF tÞ otherwise;

8>>>><>>>>: ð3:12Þ

and, as above,em�ðtÞ ¼ � infð0;emðtÞÞ:In addition,

p�t ¼ proj½0; 1�l1 � r þ ðl2 � l1ÞF t

r2

� �: ð3:13Þ

Proof. If U(x) = log(x) for each m 2 D, the solution of the unconstrained problem is

pm;�t ¼

l1 � r þ ðl2 � l1ÞF t þ mðtÞr2

;

W m;�t ¼

x exp rt þR t

0 m�ðsÞds� �

H mt

;

V ðm; xÞ ¼ log xþ rT þ E

Z T

0

mðtÞ� dt�

þ E

Z T

0

1

2

l1 � r þ ðl2 � l1ÞF t þ mðtÞr

� �2

dt

" #:

ð3:14Þ

Then the process em defined by (3.12) satisfies

emðtÞ� þ 1

2

l1� rþðl2�l1ÞF tþemðtÞr

� �2

¼ infm2D

mðtÞ� þ 1

2

l1� rþðl2�l1ÞF tþ mðtÞr

� �2 !

: �

Remark 3.4. Optimal strategies for the constrained problem are projections on [0,1] ofoptimal strategies for the unconstrained problem.

Remark 3.5. In the case of the logarithmic utility function, when t is small and thus smal-ler than the change time s with high probability, Ft is close to 0; since, by hypothesis, onealso has l1�r

r2 6 0, the optimal risky asset portfolio weight is close to 0 ; after the changetime s, Ft is close to 1, and the optimal risky asset portfolio weight is close tominð1; l2�r

r2 Þ. In both cases, we approximately recover the optimal strategies of the con-strained Merton problem with drift parameters equal to l1 or l2 respectively.

Remark 3.6. Notice that, in view of (3.4), (3.5) and (3.13), p�t can be expressed in terms ofthe prices (Su, 0 6 u 6 t).


Using the explicit value (3.12) of emðtÞ, one can obtain an explicit formula for the value

function VemðxÞ corresponding to the optimal strategy.

The derivations of the optimal portfolio weights and of the expected utility of wealthfor the logarithmic investor will serve as benchmarks to compare the performancesobtained by the Model and Detect strategy and the strategy pursued by a technical ana-lyst. Indeed, we can write (see Appendix A.3) a lengthy formula for the value functionV(x) by using a result due to Yor (see Yor, 1992 or Borodin and Salminen, 2002, formula1.20.8 p.618). See Blanchet-Scalliet et al., 2005 for further details.

4. Two Model and Detect strategies

The optimal portfolio allocation strategy in the previous section supposes that the traderis allowed to continuously change his/her portfolio allocation. In this section, the trader isallowed to change his/her allocation only once. So, the trader uses an optimal detectionprocedure to decide when to rebalance his/her portfolio. In order to facilitate the compar-ison with the performance of the technical analyst’s strategy, we here assume that the traderarbitrarily sets the stock weight to p = 0 before the drift change and subsequently to 1. Wecontinue to suppose that the trader perfectly knows all the parameters of the model.

We consider two detection methods: the first one has been proposed by Karatzas(2003), and the other one has been proposed by Shiryaev (2002). The goal of these twomethods is to find a stopping rule H which detects the instant s at which the drift ofthe stock return changes. We compute the wealth of the trader who uses one of the twoModel and Detect strategies. In both cases, the trader puts all of his/her money in thebond until H, and in the stock after H, thus his wealth satisfies:

W T ¼xS0

H

SHST1ðH6T Þ þ xS0

T1ðH>TÞ:

The time H is interpreted as the ‘‘alarm’’ time, it can occur before s (in this case, it corre-sponds to a ‘‘false alarm’’), or after s. So, the amount of time by which the stopping rule Hmisses the true change of drift point s is given by jH � sj.

The main mathematical tool used to obtain these two stopping rules is the process Ft,the (conditional) probability that the (unknown) change point appeared before the run-ning time t. For each procedure, the trader decides to invest his/her wealth in the stockwhen Ft is bigger than a given quantity respectively equal to p* for the Karatzas’ method(see (4.2)) and to A* for the Shiryaev method (see (4.3)). These two quantities will bedefined in Sections 4.1 and 4.2.

With the Karatzas’ method, the trader minimizes the expected miss EjH� sj. With Shir-yayev’s one, the trader minimizes fPðH < sÞ þ cEðH� sÞþg, i.e., he/she does not give thesame weight to errors generated by false alarms (H < s) and errors generated by latealarms (H P s ).

4.1. Karatzas’ method

We first adapt Karatzas’s detection method to compute the optimal stopping rule HK

that minimizes the expected miss

RðHÞ :¼ EjH� sj ð4:1Þ


over all stopping rules H, when s is assumed to have the a priori exponential distribution(2.2).

Proposition 4.1. The stopping rule HK which minimizes the expected miss EjH� sj over all

the stopping rules H with EðHÞ <1 is

HK ¼ infft P 0 F t P p�j g; ð4:2Þwhere Ft is defined as in (3.5) and p* is the unique solution in ð1

2; 1Þ of the equationZ 1=2

0

ð1� 2sÞe�b=s

ð1� sÞ2þb s2�b ds ¼Z p�

1=2

ð2s� 1Þe�b=s

ð1� sÞ2þb s2�b ds;

where b = 2kr2/(l2 � l1)2.

Proof. We adapt Karatzas’s method in Karatzas (2003) to our specific case. Denote by Sthe collection of stopping rules H : X! [0,1) such that EðHÞ <1. Rewrite (4.1) asfollows:

RðHÞ ¼ E½ðH� sÞþ þ ðs�HÞþ� ¼ EðH� sÞþ þ EðsÞ � EðH ^ sÞ:Then, using (2.2) and the notation (3.5), we get

RðHÞ ¼ 1

kþ E

Z H

0

1ðs6sÞ ds�Z H

0

1ðs>sÞ ds� �

¼ 1

kþ E

Z 1

0

ð21ðs6sÞ � 1Þ1ðs6HÞ ds� �

¼ 1

kþ 2E

Z H

0

F s �1

2

� �ds:

We thus obtain

infH2S

RðHÞ ¼ 1

kþ 2 inf

H2SE

Z H

0

F s �1

2

� �ds:

It now remains to follow Karatzas’ arguments (see Karatzas, 2003). h

The terminal wealth of a trader who uses Karatzas detection procedure satisfies

W T ¼ xerHK

exp rðBT � BHKÞ þ l1 �r2

2

� �ðT �HKÞ

�þðl2 � l1Þ½ðT � sÞþ � ðHK � sÞþ�

�1ðHK

6T Þ þ x expðrT Þ1ðHK>T Þ:

4.2. Shiryaev method

The detection method proposed by Shiryaev (namely the Variant B in Shiryaev (2002))consists in computing

BðcÞ :¼ infHfPðH < sÞ þ cEðH� sÞþg

and the corresponding optimal stopping time for a given c > 0.


For all c > 0, this time is given by

HSðA�Þ :¼ infft P 0; F t P A�g ð4:3Þwhere F is the conditional a posteriori probability solution of (A.6) and the parameter A*

is defined as the root in (0,1) of the equationZ A�

0

exp � 2kr2

ðl2 � l1Þ2

1

y

!1

yð1� yÞ2y

1� y

� � 2kr2

ðl2�l1Þ2

dy

¼ ðl2 � l1Þ2

2r2cexp � 2kr2

ðl2 � l1Þ2

1

A�

!A�

1� A�

� � 2kr2

ðl2�l1Þ2

:

For both Karatzas and Shiryaev’s detection procedures, we are able to write explicitformulae for the expected utility of terminal wealth, EðlogðW TÞÞ (similar to the formulain Proposition 6.1 below). Due to their complexity, these formulae do not allow us to com-pare the expected utility of terminal wealth associated with the Model and Detect strate-gies to the one obtained when pursuing the optimal strategy or the technical analysis basedstrategy. In Section 7, we therefore rely on numerical simulations to make comparisonsbetween the various strategies’ performances.

Remark 4.2. Beibel and Lerche (1997) have considered the model (2.1) with l1 � r2

2 Pr P l2 � r2

2 . They have studied the problem of maximizing Eðe�rHSH1ðH<1ÞÞ over allstopping times H adapted to the filtration generated by (St). We do not examine theirdetection procedure in our study.

5. Mis-specified trading models

In reality, it is extremely difficult to know the parameters characterizing the investmentopportunity set exactly. It may be possible to calibrate the first drift coefficient l1 and thevolatility coefficient r relatively well owing to historical data, but the value of l2 cannot bedetermined a priori (i.e. before the occurrence of the drift change), and the law of s cannotbe calibrated accurately because of the lack of data associated with s. It is thus reasonableto assess the impact of estimation risk on the performance of the various model-baseddetection strategies.

5.1. Mis-specification of the parameters

We consider the case where each parameter is estimated with error. In other words, thetrader believes that the stock price satisfies the following stochastic differential equation:

dSt ¼ Stð�l2 þ ð�l1 � �l2Þ1ðt6�sÞÞdt þ �rSt dBt; ð5:1Þ

where the law of �s is exponential with parameter �k, while the true stock price is still givenby (2.1). Our aim is to study the mis-specified optimal allocation strategy and the mis-spec-ified Model and Detect strategy.

Notation. As in the previous developments without estimation error, St denotes thecurrent stock price. We need to compute Lt and Ft (see (3.4) and (3.5)) to apply the optimal


allocation strategy and the Model and Detect strategies. We define the approximatedquantities computed when the model is mis-specified Lt and F t as follows:

Lt ¼ exp1

�r2ð�l2 � �l1Þ logðStÞ �

1

2�r2ð�l2 � �l1Þ2 þ 2ð �l2 � �l1Þ �l1 �

�r2

2

� �� t

� �;

F t ¼�ke

�ktLt

R t0

e��ksL�1

s ds

1þ �ke�ktLt

R t0

e��ksL�1s ds

:

5.2. On the mis-specified optimal allocation strategy

Observing the stock price St, the trader computes a ‘‘pseudo optimal’’ portfolio alloca-tion by using the erroneous parameters �l1 , �l2, �r and �k. Thus, the stock proportion of his/her mis-specified optimal allocation strategy satisfies

�p�t ¼ proj½0; 1��l1 � r þ ð�l2 � �l1ÞF t

�r2;

and the corresponding wealth satisfies

W �t ¼ ert exp

Z t

0

�p�udðe�ruSuÞ� �

:

5.3. On mis-specified Model and Detect strategies

The erroneous stopping rule for the Karatzas detection time rule satisfies

HK ¼ infft P 0; F t P p�gwhere �p� is the unique solution in ð1

2; 1Þ of the equationZ 1=2

0

ð1� 2sÞe��b=s

ð1� sÞ2þ�bs2��b ds ¼

Z �p�

1=2

ð2s� 1Þe��b=s

ð1� sÞ2þ�bs2��b ds

with �b ¼ 2�k�r2=ð�l2 � �l1Þ2.The wealth of the Model and Detect trading strategy satisfies

W T ¼ xS0HK

ST

SHK

1ðHK6T Þ þ xS0T1ðHK>T Þ:

A similar approach based on the results obtained in Section 4.2 can be followed tocompute the erroneous stopping rule and its associated wealth for the Shiryaev detectionrule.

6. The chartist investment strategy

6.1. Introduction

Technical analysis makes predictions about the evolution of an asset’s price anddefines trading rules using only the asset’s price (or/and volume) history. Thus, technicalanalysts compute indicators based on the asset’s past transaction prices and volumes.


These indicators are used as trading signals assuming that (see, e.g., the book by Achelis,2000):

• The price of a stock is governed by the law of supply and demand.• The stock price evolves according to trends during discernible periods.• These discernible tendencies repeat themselves in a regular fashion.

A very large number of technical analysis indicators are used by practitioners. Intheir impressive study, Sullivan et al. (1999) provide a parameterization for 7846 distincttrading rules. Here, we limit ourselves to the simple moving average indicator because it iseasy to compute and widely used to detect trend patterns in stock prices. In order tocompute its value, one averages the closing prices of the stock during the d most recenttime periods. When prices are trending, this indicator reacts quickly to recent pricechanges.

6.2. Moving average indicator for the stock prices

Consider a chartist trader who takes decisions at discrete times during the interval [0,T]with time increments Dt ¼ T

N:

0 ¼ t0 < t1 < . . . < tN ¼ T ; tk ¼ kDt:

We denote by pt 2 {0,1} the proportion of the agent’s wealth invested in the risky asset attime t, and by Md

t the simple moving average indicator of the prices defined as

Mdt ¼

1

d

Z t

t�dSu du: ð6:1Þ

The parameter d denotes the size of the time window used to compute the moving average.At time 0, the agent knows the past prices of the stock and has enough data to com-

pute Md0. At each tn, n 2 [1 � � � N], the chartist follows a very simple trading strategy: all

the wealth is invested into the risky asset if the price Stn is larger than the moving aver-age Md

tn. Otherwise, all the wealth is invested into the riskless asset. This portfolio

investment strategy is thus analogous to the one followed by the Model and Detecttrader.

Consequently,

ptn ¼ 1ðStn PMdtnÞ: ð6:2Þ

Denote by x the initial wealth of the trader. The wealth at time tn+1 is

W tnþ1¼ W tn

Stnþ1

Stn

ptn þS0

tnþ1

S0tn

ð1� ptnÞ !

;

and therefore, since S0tnþ1=S0

tn¼ expðrDtÞ,

W T ¼ xYN�1

n¼0

ptn

Stnþ1

Stn

� expðrDtÞ� �

þ expðrDtÞ�

: ð6:3Þ


6.3. The particular case of the logarithmic utility function

We now assume that the chartist trader displays a logarithmic utility function. Then, hisexpected utility of wealth can be explicitly characterized.

Proposition 6.1. Consider a technical analyst whose strategy is defined as in (6.2). Then his

expected logarithmic utility of wealth satisfies

E logW Te�rT

x

� �� ¼ l2 �

r2

2� r

� �Tpð1Þd

þ Dt l2 �r2

2� r

� �1� e�kT

1� e�kDtðpð2Þd � pð1Þd Þekd þ pð3Þd

� �� Dtðl2 � l1Þðe�kDt � kDtÞ 1� e�kT

1� e�kDtpð3Þd ;

where we have set

pð1Þd ¼Z 1

0

Z 1

y

zl2�3=2

2ye� l2=r�r=2ð Þ2d

2 �ð1þz2Þ2r2y ir2d=2

zr2y

� �dzdy;

pð2Þd ¼Z d

0

ZR4

1dy2P

z1y1þz2

n o zl2�3=22

2y2

e� l2=r�r=2ð Þ2ðd�vÞ

2 �ð1þz2

2Þ

2r2y2 ir2ðd�vÞ=2

z2

r2y2

� �

� zl1�3=21

2y1

e� l1=r�r=2ð Þ2v

2 �ð1þz2

1Þ

2r2y1 ir2v=2

z1

r2y1

� �e�kv dy1 dz1 dy2 dz2 dv;

pð3Þd ¼Z 1

0

Z 1

y

zl1�3=2

2ye� l1=r�r=2ð Þ2d

2 �ð1þz2Þ2r2y ir2d=2

zr2y

� �dzdy;

the function i being defined as in (A.11).

Proof. In view of

Stjþ1

Stj

¼ exp l1 �r2

2

� �Dt þ ðl2 � l1Þ tjþ1 �maxðs; tjÞ

� �þ þ rðBtjþ1� BtjÞ

� �; ð6:4Þ

and (6.2), we have

E logW T

x

� �¼ l1 �

r2

2� r

� �DtXN�1

j¼0

P Stj P Mdtj

� �

þ ðl2 � l1ÞXN�1

j¼0

E 1ðStj PMdtjÞ tjþ1 �maxðs; tjÞ� �þn o

þ rT þ rXN�1

j¼0

E 1ðStj PMdtjÞðBtjþ1

� BtjÞn o

:

From here, one obtains the result by lengthy calculations using independence and identityin law arguments, and Lemma A.3 in the Appendix. h


Remark 6.2. Relying on Proposition 6.1, one can use numerical optimization proceduresto optimize the choice of d, the moving average window size. As we will see in the nextsection, inadequate choices of d may negatively affect the performance of the technicalanalyst strategy.

7. A numerical comparison of the various strategies

7.1. Empirical determination of a good windowing

Before turning to a comparison between the various trading strategies, we first showhow to optimize the choice of the moving average window size d by using Proposition6.1 and deterministic numerical optimization procedures, or by means of Monte Carlosimulations. In this subsection, we present results obtained from Monte Carlo simulationsto show that inadequate choices of d may indeed alter the performance of the technicalanalyst strategy. For each value of d we have simulated 500,000 trajectories of the assetprice and computed the expected utility of terminal wealth, E log ðW TÞ by a Monte Carlomethod. In all our simulations the empirical variance of log(WT) is set at 0.04. Thus, theMonte Carlo error on E logðW TÞ is of the order 5.10�4 with probability 0.99. At first, thenumber of price trajectories used for these simulations may seem too large; however, con-sidered as a function of d, the quantity E logðW TÞ varies very slowly, so that we really needa large number of simulations to obtain smooth curves (cf. Fig. 1).

Fig. 1a and b illustrates the relationship between E logðW TÞ and d for two different val-ues of the stock returns’ volatility. It is clear from these figures that the optimal choice of dvaries with the volatility of the stock returns. When the volatility reaches 0.05, the optimalchoice of d is around 0.3 whereas, when the volatility increases to 0.15, the optimal choiceof d is around 0.8.

On the basis of a comprehensive numerical study performed in Blanchet-Scalliet et al.(2005), we can make the following statements:

• When the volatility decreases, the choice of d becomes more important: the curve is flatfor large volatility levels only.– When the volatility is high, the losses decrease when choosing a large window size.– In all cases, a too small window length is sub-optimal,

4.72

4.73

4.74

4.75

4.76

4.77

4.78

4.79

4.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

E(l

og(W

_T))

order of moving average = delta

4.77

4.78

4.79

4.8

4.81

4.82

4.83

4.84

4.85

4.86

4.87

4.88

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

E(l

og(W

_T))

order of moving average = delta

Fig. 1. Expected values of the logarithm of the terminal wealth (EðlogðW TÞÞ) for a trader using the movingaverage indicator of order d as a function of d. (a) l1 = �0.2; k = 2; r = 0.0; l2 = 0.2; r = 0.15; T = 2.0.(b) l1 = �0.2; k = 2; r = 0.0; l2 = 0.2; r = 0.05; T = 2.0.

0 0.5 1 1.5 24.63

4.64

4.65

4.66

T=1

0 0.5 1 1.5 24.92

4.94

4.96

4.98

5T=3

Fig. 2. Expected values of the logarithm of the terminal wealth (EðlogðW T ÞÞ) for a trader using the movingaverage indicator of order d as a function of d: the dependence of the optimal d on the time horizon T.

2

fou


– As jl1j or l2 decreases (respectively, increases), the choice of d becomes less (respec-tively, more) important.

• The choice of d depends on the arrival rate k.• The parameter l2 has a strong effect on the importance of the window length. In Blan-

chet-Scalliet et al. (2005), it is shown that the curves become flatter when l2 increases.This observation confirms the intuition; if the future drift is not large enough, the detec-tion of s will be more difficult.

Fig. 2 shows that, when k = 2.0, the time horizon has a significant effect on the optimalchoice of d. When the time horizon is small, for example when T = 1, one has better tounderestimate d than to overestimate it. When T is large, one has better to overestimated. Of course, the specific values for T highly depend on the chosen level of k.

7.2. Mis-specified mathematical strategies vs technical analysis

We can now address our main question: Is it better to invest according to a mathemat-ical Model and Detect strategy based on a mis-specified model, or according to a strategywhich is model – free? Due to the analytical complexity of all the explicit formulae that wehave obtained for the various expected utilities of terminal wealth, we have not yet suc-ceeded to find a mathematical answer to this question (even in asymptotic cases, whenl2�l1

r2 is large, e.g.). We therefore present numerical results obtained from Monte Carlo sim-ulations to illustrate our comparisons.2

Fig. 3 shows that, despite a large degree of parameter mis-specification, the two modeland detect strategies represented in panels (a) and (b) yield a good performance and clearlyout-perform the technical analyst. However, it can be seen from Fig. 3c that the optimaltrading strategy with mis-specified parameters is out-performed by the technical analysisstrategy. This result suggests that statistical detection techniques or technical analysisapproaches could be more attractive when the parameters are mis-specified.

We have run, a number of other simulations which all confirm that the technical analystmay out-perform the mis-specified optimal allocation strategy but not the mis-specified

A more complete set of simulations on the performance of the mathematical and the chartist strategies can bend in Blanchet-Scalliet et al. (2005).

0 0.5 1 1.5 24.55

4.6

4.65

4.7

4.75

4.8

4.85

Time (years)

technical analysisexact parametersmisspecified parameters

0 0.5 1 1.5 24.55

4.6

4.65

4.7

4.75

4.8

4.85

E[lo

g(W

T)]

E[lo

g(W

T)]

E[lo

g(W

T)]

Time (years)


0 0.5 1 1.5 24.55

4.6

4.65

4.7

4.75

4.8

4.85

Time (years)


μ = −0.2μ = 0.2σ = 0.15λ = 2

21

exactparameters

2

parametersmisspecified

μ = −0.3

λ = 1

_

___1μ = 0.1σ = 0.25

Fig. 3. Comparison of the expected values of the logarithm of wealth when the traders use strategies with exactand mis-specified parameters. (a) Model and Detect strategy based on Karatzas’ estimation, (b) Model and Detectstrategy based on Shiryaev’s estimation and (c) optimal strategy.


Model and Detect strategies. These simulations also show that when l2/l1 decreases, theperformances of well specified and mis-specified Model and Detect strategies decrease.3

In conclusion, our numerical study suggests that there is no universal solution to theproblem of parameter mis-specification. It seems that when the drifts are high in absoluteterms and, in particular, when the upward drift is high, the performance of the Model andDetect strategies can be quite robust and superior to the one of the chartist trading strat-egy. However, their performance deteriorates rapidly when k is strongly mis-specified and/or when the upward drift is not very high. Since the second drift is in fact the hardest toestimate due to the fact that we lack a priori information, we recommend caution beforeasserting that Model and Detect strategies are superior to the technical trading rule.4

Indeed, the Model and Detect strategies only offer a clear comparative advantage overthe chartist trading rule in the presence of strong expected future trends.

3 The results are available from the authors upon request.4 In reality, the technical analyst does not know the length of the optimal window, thus his/her strategy is not

free of mis-specification either. However, as we show in Fig. 2, the performance of the technical analyst has onlyweak sensitivity with respect to the window length.


8. Conclusion and perspectives

In this study, we have compared the performance of trading strategies based on possi-bly mis-specified mathematical models used to detect the time of the change in the driftof the stock return with a trading strategy based on the simple moving average rule.We have explicitly computed the trader’s expected logarithmic utility of wealth for thevarious trading strategies. Unfortunately, these explicit formulae were not propitiousto mathematical comparisons. We have therefore relied on Monte Carlo numericalexperiments, and observed from these experiments that under parameter mis-specification,the technical analysis technique out-performs the optimal allocation strategy but notthe model and detect strategies. The latter strategies dominance is confirmed underparameter mis-specification as long as the two stock returns’ drifts are high in absoluteterms.

This study provides a first step towards building a rigorous mathematical framework inwhich chartist and mathematical model based trading strategies can be compared. We areextending this research along several dimensions. First, we examine and model the perfor-mance of other chartist based trading rules (such as filter rules, point and figure charts,etc.). Second, we consider modeling the more realistic case where there are multiplechanges in the drift of the stock returns: we examine the case where the instantaneousexpected rate of return of the stock changes at the jump times of a Poisson’s process,and the value of this rate after each time change is unknown. We follow two new direc-tions to tackle these questions: jointly with B. de Saporta (INRIA), we use stochastic con-trol techniques for switching models and, jointly with M. Martinez (INRIA) andS. Rubenthaler (University of Nice Sophia Antipolis), we use filtering techniques (Marti-nez et al., 2006). Finally, it would be worth extending our conceptual framework to themore realistic case where the mathematical and the chartist strategies’ performancesaccount for market frictions.

Acknowledgement

Financial support by the National Centre of Competence in Research ‘‘Financial val-uation and Risk Management’’ (NCCR FINRISK) is gratefully acknowledged. NCCR-FINRISK is a research program supported by the Swiss National Science Foundation.Awa Diop gratefully acknowledges financial support from the National Centre of Compe-tence in Research ‘‘Conceptual Issues in Financial Risk Management’’ (NCCR–FINRISK).

Appendix A

A.1. An explicit expression for the process Ft

Lemma A.1. The conditional a posteriori probability Ft that the change point has appearedbefore time t, that is, F t :¼ Pðs 6 tjFS

t Þ, is

F t ¼kektLt

R t0

e�ksL�1s ds

1þ kektLt

R t0

e�ksL�1s ds

ðA:1Þ


where

Lt ¼St

S0

� �l2�l1r2

exp � 1

2r2ðl2 � l1Þ

2 þ 2ðl2 � l1Þ l1 �r2

2

� �� t

� �: ðA:2Þ

Proof. In order to compute Ft, we start with a probability space ðX;F;QÞ that can sup-port both a standard Brownian motion eB and independent random variable s: X! [0,1)with distribution Q½s > t� ¼ e�kt for t P 0; we denote by ðGtÞtP0 the filtration generated bys and ~B, which means that Gt :¼ rðs; eBs; 0 6 s 6 tÞ. In view of Girsanov’s theorem, theprocess

Bt ¼ eBt �ðl2 � l1Þ

r

Z t

0

1ðs6sÞ ds

is a ðGtÞtP0-Brownian motion under the measure of probability P such that

dP

dQ

Gt

¼ exp

Z t

0

1

rðl2 � l1Þ1ðs6sÞ deBs �

1

2r2

Z t

0

ðl2 � l1Þ21ðs6sÞ ds

� �¼ exp

1

rðl2 � l1ÞðeBt � eBsÞ1ðs6tÞ �

1

2r2ðl2 � l1Þ

2ðt � sÞþ� �

¼ exp1

rðl2 � l1ÞðeBt � eBsÞ �

1

2r2ðl2 � l1Þ

2ðt � sÞ� �

1ðs6tÞ þ 1ðs>tÞ:

We observe that

eBt ¼1

rlog

St

S0

� �� l1 �

r2

2

� �t

� �: ðA:3Þ

Observe that under Q, the random variable s is independent of the ðGt; t P 0Þ-Brownianmotion eB and thus of R. Thus, the above expression of dP=dQ can be written as follows:

dP

dQ

Gt

¼ Lt

Ls1ðs6tÞ þ 1ðs>tÞ :¼ Zt; ðA:4Þ

where Lt is defined as in (3.4). We now check that, on the probability space ðX;F;PÞ, wehave the same model as the one presented in the introduction. Indeed, the random variables is G0 -measurable; then, under P, s is independent of the ðGtÞtP0-Brownian motion B andwe have P½s > t� ¼ EQ½Z01s>t� ¼ Q½s > t�. Using the Bayes rule, one gets

F t ¼ P½s 6 tjFSt � ¼

EQ Zt1ðs6tÞjFSt

�EQ ZtjFS

t

� : ðA:5Þ

Using now the independence of eB and s under Q, one gets

EQ ZtjFSt

�¼ EQ

Lt

Ls1ðs6tÞ þ 1ðs>tÞjFS

t

� ¼Z t

0

Lt

Lske�ks dsþ e�kt:

On the other hand, we get

EQ Zt1ðs6tÞjFSt

�¼Z t

0

Lt

Lske�ks ds:

Then, going back to (A.5), we get (3.5).


Moreover, we can show that the process (Ft)tP0 satisfies the following stochasticdifferential equation

dF t ¼ kð1� F tÞdt þ ðl2 � l1Þr

F tð1� F tÞdBt; ðA:6Þ

where

Bt ¼1

rlogðStÞ � l1 �

r2

2

� �t � ðl2 � l1Þ

Z t

0

F s ds� �

; t P 0; ðA:7Þ

is the innovation process. Indeed, set V t :¼ F t1�F t

; an easy computation leads to

V t ¼Z t

0

kLt

Lsekðt�sÞ ds:

From

dLt ¼ Lt1

r2ðl2 � l1Þdðlog StÞ �

1

r2ðl2 � l1Þ l1 �

r2

2

� �dt

� �we deduce

dV t ¼ kþ V t k� 1

r2ðl2 � l1Þ l1 �

r2

2

� �� dt þ 1

r2ðl2 � l1ÞV t dðlog StÞ;

V 0 ¼ 0:

We finally apply Ito’s formula to the process F t ¼ V t1þV t

, and get the stochastic differential

Eq. (A.6). Notice that Bt defined as in (3.6) is a ðFSt ÞtP0-Brownian motion. h

We conclude with a result useful to apply the martingale representation theorem in thenext subsection.

Lemma A.2. The filtration generated by the observations ðFS� Þ is equal to the filtration

generated by the innovation process ðBt; t P 0Þ. In particular, each ðFS� Þ martingale M

admits a representation as

Mt ¼ M0 þZ t

0

/s dBs;

where / is an ðFS� Þ adapted process.

Proof. Thanks to (3.6), Bt is ðFS� Þ adapted. Conversely, we write (3.6) as:

dðlog StÞ ¼ l1 �r2

2

� �þ ðl2 � l1ÞF t

� �dt þ rdBt; ðA:8Þ

Thanks to (A.6), F is ðFB� Þ adapted. and we conclude that the process SÆ(=exp(RÆ)) is also

ðFB� Þ adapted. And so ðFB

� Þ ¼ ðFS� Þ. h

A.2. Proof of proposition 3.1

We follow Karatzas’s method (see for example Karatzas, 1997). For p 2Aðx; mÞ,remember that


dW m;pt

W m;pt

¼ ðr þ m�ðtÞÞdt þ pt ðl1 � rÞ þ ðl2 � l1ÞF t þ mðtÞð Þdt þ ptrdBt:

Define

ct :¼ ðl1 � rÞ þ ðu2 � l1ÞF t þ mðtÞ

W m;pt :¼ W m;p

t exp �rt �Z t

0

m�ðsÞds� �

:

We thus have

dW m;pt

W m;pt

¼ ptct dt þ ptrdBt;

W m;pt ¼ x exp

Z t

0

pscs �1

2r2p2

s

� �dsþ

Z t

0

rps dBs

� �:

We now search an exponential martingale Mt (independent of p) such that eW m;pt ¼ W m;p

t Mt

is an exponential martingale. Set

Mt ¼ exp

Z t

0

/sdBs �1

2

Z t

0

/2s ds

� �:

Then / needs to satisfy

� 1

2ð/s þ psrÞ2 ¼ pscs �

1

2/2

s �1

2r2p2

s ;

from which /s ¼ � csr and Mt ¼ H m

t . Thus, for all p,

d eW m;pteW m;p

t

¼ rpt �ct

r

� �dBt: ðA:9Þ

Therefore the process ð eW m;pt ; 0 6 t 6 T Þ is a non-negative ðFS

t ;PÞ-local martingale and soa supermartingale. Consequently,

E H mTW m;p

T exp �rT �Z T

0


6 x:

We now introduce the convex dual of U(Æ):eU ðyÞ :¼ max0<x<1

½UðxÞ � xy�; y > 0:

Using a duality method, we obtain: for all m > 0, p 2Aðx; mÞ, y P 0,

E½UðW m;pT Þ� 6 E eU yH m

T exp �rT �Z T

0


þ yE H mTW m;p

T exp �rT �Z T

0


6 E eU yH mT exp �rT �

Z T

0


þ yx:


This inequality is an equality if and only if

W m;pT ¼ ðU 0Þ

�1 yH mT expð�rT �

R T0 m�ðtÞdtÞ

� �;

E H mTW m;p

T exp �rT �R T

0m�ðtÞdt

� �h i¼ x:

8><>:The coefficient y is introduced to satisfy the constraint.

Now, we need to verify that there exists a portfolio such that the process

X t :¼E H m

T expð�rT �R T

0m�ðsÞdsÞðU 0Þ�1 yH m

T expð�rT �R T

0m�ðsÞdsÞ

� �FS

t

�hH m

t expð�rt �R t

0m�ðsÞdsÞ

is its wealth process. We use the martingale representation property of the Brownian fil-tration in order to find the optimal strategy p*. Indeed, there exists a predictable process/ such that

E H mTe�rT�

R T

0m�ðtÞ dtðU 0Þ�1 yH m

Te�rT�

R T

0m�ðtÞ dt

� �=FS

t

� �¼ xþ

Z t

0

/s dBs:

In particular, with the notation eX t ¼ X tH mt expð�rt �

R t0 m�ðsÞdsÞ, we obtain

deX t ¼ /t dBt:

Consider the strategy

p�t ¼ r�1 l1 � r þ ðl2 � l1ÞF t þ mðtÞr

þ /t

X tH mt e�rt�R t

0m�ðuÞdu

0@ 1A:In view of (A.9), we have

d eW m;p�teW m;p�

t

¼ /teX t

dBt ¼deX teX t

:

By uniqueness arguments, we obtain X t ¼ W m;p�

t .

A.3. Joint law of geometric Brownian motion and its integral

Lemma A.3. Let B be a real Brownian motion. Let r > 0 and m be in R. Let G be a geometric

Brownian motion:

Gs ¼ expðr2msþ rBsÞ:It holds that

P

Z t

0

Gs ds 2 dy; Gt 2 dz� �

¼ zm�1

2ye�m2r2 t

2 �ð1þz2Þ2r2y ir2 t

2

zr2y

� �dy dz; ðA:10Þ

where

iyðzÞ :¼ zep2=4y

pffiffiffiffiffipyp

Z 1

0

exp �z cosh u� u2=4y� �

sinh u sinðpu=2yÞdu: ðA:11Þ

Yor has obtained this last result in Yor (1992) (see also Borodin and Salminen, 2002).


References

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